Numerical Methods for the QCD Overlap Operator: II. Optimal Krylov SubspaceMethods (original) (raw)

Numerical methods for the QCD overlap operator: III. Nested iterations

Computer Physics Communications, 2005

The numerical and computational aspects of chiral fermions in lattice quantum chromodynamics are extremely demanding. In the overlap framework, the computation of the fermion propagator leads to a nested iteration where the matrix vector multiplications in each step of an outer iteration have to be accomplished by an inner iteration; the latter approximates the product of the sign function of the hermitian Wilson fermion matrix with a vector.

Numerical methods for the QCD overlap operator IV: Hybrid Monte Carlo

Computer Physics Communications, 2009

The computational costs of calculating the matrix sign function of the overlap operator together with fundamental numerical problems related to the discontinuity of the sign function in the kernel eigenvalues are the major obstacles towards simulations with dynamical overlap fermions using the Hybrid Monte Carlo algorithm. In a previous paper of the present series we introduced optimal numerical approximation of the sign function and have developed highly advanced preconditioning and relaxation techniques which speed up the the inversion of the overlap operator by nearly an order of magnitude.

Two-grid overlap solver in lattice QCD

Proceedings of The 38th International Symposium on Lattice Field Theory — PoS(LATTICE2021)

In lattice quantum chromodynamics with chiral fermions we want to solve linear systems which are chiral and dense discretizations of the Dirac operator, or the overlap operator. We propose that multigrid solver are the best choice to solve quickly this linear systems. In this paper we develop a two-grid algorithm. For this purpose, we use the equivalence of the overlap operator with the truncated overlap operator, which is a five dimensional formulation of the same theory. The coarsening is performed along the fifth dimension only. We have tested first this algorithm for small lattice volume 8^4 and we bring here our results for larger lattice size 16^4. We have done simulation in the range of coupling constants and quark masses for which the algorithm is fast and saves a factor of 6, even for dense lattice, compared to the standard Krylov subspace methods.

A numerical method for perturbative QCD calculations

2004

Standard methods for performing analytic perturbative calculations for the process of e+ e -+ qq up to 0(a3 ) are explained and the results given. An emphasis is given to the organisation of calculations using the Cutkosky cutting rules and the renormalisation of the massive quark propagator. Methods for numerical integration are presented including those used in VEGAS. The numerical methods used in the Beowulf program for calculating infra-red safe observables for jet events from electron-positron collisions are also explained. Cancellations of singularities required for numerical calculations are demonstrated using an example in 03 theory both numerically and graphically. Renormalisation by subtraction of appropriate integrals is also covered. Adaptations of the Beowulf procedure required for the inclusion of massive fermions are developed and explained. An alternative method for including the quark self energy and its related cuts using scalar decomposition, numerically equivalen...

Multigrid preconditioning for the overlap operator in lattice QCD

Numerische Mathematik, 2015

The overlap operator is a lattice discretization of the Dirac operator of quantum chromodynamics, the fundamental physical theory of the strong interaction between the quarks. As opposed to other discretizations it preserves the important physical property of chiral symmetry, at the expense of requiring much more effort when solving systems with this operator. We present a preconditioning technique based on another lattice discretization, the Wilson-Dirac operator. The mathematical analysis precisely describes the effect of this preconditioning in the case that the Wilson-Dirac operator is normal. Although this is not exactly the case in realistic settings, we show that current smearing techniques indeed drive the Wilson-Dirac operator towards normality, thus providing a motivation why our preconditioner works well in computational practice. Results of numerical experiments in physically relevant settings show that our preconditioning yields accelerations of up to one order of magnitude.

Considerations on numerical analysis of QCD

1983

Nuclear Physics B225 [FS9] (1983) 475-496 North-Holland Publishing Company CONSIDERATIONS ON NUMERICAL ANALYSIS OF QCD HW HAMBER The Institute for Advanced Studies, Princeton, NJ08540, USA E. MARINARI Service de Physique Thrique, CEN-SACLAY, 91191 Gif sur Yvette, France G. PARISI Universitdi Roma ll, Tor Vergata, Roma, Italy C.

On the numerical solution of the evolution equation in QCD

Czechoslovak Journal of Physics, 1986

We discuss in some detail the algorithm for the numerical solution of QCD evolution equations in the next-to-leading order. Relative merits and shortcomings of this method are presented and compared with those of other available approaches. Effective control of the accuracy of obtained results is stressed.

Block Algorithms for Quark Propagator Calculation

1997

Computing quark propagators in lattice QCD is equivalent to solving large, sparse linear systems with multiple right-hand sides. Block algorithms attempt to accelerate the convergence of iterative Krylov-subspace methods by solving the multiple systems simultaneously. This paper compares a block generalisation of the quasi-minimal residual method (QMR), Block Conjugate Gradient on the normal equation, Block Lanczos and (γ_5-symmetric) Block BiConjugate Gradient.

Algorithms in lattice QCD

1998

The enormous computing resources that large-scale simulations in Lattice QCD require will continue to test the limits of even the largest supercomputers into the foreseeable future. The efficiency of such simulations will therefore concern practitioners of lattice QCD for some time to come. I begin with an introduction to those aspects of lattice QCD essential to the remainder of the thesis, and follow with a description of the Wilson fermion matrix M, an object which is central to my theme. The principal bottleneck in Lattice QCD simulations is the solution of linear systems involving M, and this topic is treated in depth. I compare some of the more popular iterative methods, including Minimal Residual, Corij ugate Gradient on the Normal Equation, BI-Conjugate Gradient, QMR., BiCGSTAB and BiCGSTAB2, and then turn to a study of block algorithms, a special class of iterative solvers for systems with multiple right-hand sides. Included in this study are two block algorithms which had ...